Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

摘要

In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.